Aircraft: PA-44 SEMINOLE/TURBO SEMINOLE
Phase of flight: Cruise
Anomaly: Conflict NMAC

While conducting multi-engine flight training with one engine shut down; flying eastbound in level flight at 4;500; my student and I received a traffic advisory from Oklahoma City Approach Control referencing a Gulfstream to our east; descending to 4;000. The student was practicing simulated instrument conditions with his outside visibility restricted and I was acting as safety pilot watching for traffic. I told the student that with existing visibility and ceilings it might be difficult to see traffic from that direction and instructed him to turn to the west; away from the potential traffic conflict. The Gulfstream traffic also received a traffic call referencing our position and after a few moments replied 'Seminole in sight.' Due to our new westbound heading; traffic arriving into the terminal area from the east would have been obscured by our fuselage and not visible to us. About one minute later The Gulfstream appeared on our right approximately 2;000 horizontal feet north of us and descending through our altitude in a 30 degree bank from west to south turning towards our aircraft on what appeared to be a near collision course. The Gulfstream then appeared to increase its rate of descent allowing it to pass approximately 400 FT directly under our aircraft as if the Gulfstream crew had only seen our aircraft at the last moment. Due to the high speed of the Gulfstream and the sudden nature of the incident I did not have time to react before the Gulfstream had already passed beneath us. After passing beneath us the Gulfstream appeared to continue its descent to perhaps as low as 3;000 FT MSL as it recovered from the high rate of descent required to avoid us. I heard the Gulfstream crew advise the controller that they had descended to 3;500 to avoid traffic.It's not clear to me how the situation came to occur. I do not know if the Gulfstream was vectored into us or if the Gulfstream was providing its own navigation. After the incident I asked the Approach Controller 'Oke City Approach; [call sign] I thought that Gulfstream said they had us in sight' and the Gulfstream crew replied 'Guess we got a real good look at you'. I understood the statement to mean they also thought we had gotten close. If the controller had advised the Gulfstream to stop above us or issued the Gulfstream a vector to avoid us the potential hazard would have been avoided. Also; the Gulfstream crew presumably lost visual contact with us and had they notified the controller of that fact he could have responded appropriately. While conducting single engine operations in simulated IMC during training for the Multi Engine Commercial rating we experienced a near miss with other traffic. While flying heading 090 at 4;500 FT MSL Approach called; '(Tail number); Gulfstream traffic 12 o'clock; three miles; southbound; out of 5;000 FT to which I replied 'Searching; (last three of tail number)'. They then called the Gulfstream stating our position to which the crew responded; 'In sight'. After about 20 seconds I elected to make a 180 degree turn to the right to fly heading 270 because my instructor had not made visual contact with the traffic and we were not sure if the Gulfstream was VMC even though they had called us 'in sight' based on estimated ceiling height above us. About a minute later while flying heading 270; assuming we had turned away from the traffic; my instructor showed extreme surprise and indicated that the Gulfstream had passed right to left 'just below us' and had entered a dive shortly before reaching our lateral position causing a loss of separation/near miss; missing by an estimated 300 FT. I came out from under the hood; looked south; and saw the Gulfstream traffic below and (now) south leveling out from an apparent quick descent. My instructor called approach stating; 'Approach; (tail number); I thought that the Gulfstream traffic had us in sight' to which approach replied; 'Affirmative'. The Gulfstream crew keyed up stating; 'Yeah; we got a real good look at ya'! A few moments later approach called a B737 in our area requesting a report on cloud bases which were reported as 5;700 FT MSL.It is unclear how; after our turn to fly heading 270; the Gulfstream intercepted our flight path when it was originally reported as '12 o'clock; three miles; southbound' on our original heading of 090. I do not know for sure but contributing factors could be a flight crew's over reliance on TCAS painting targets without visually verifying traffic; traffic scanning procedures; and approach's work load causing them to miss a possible loss of separation despite the other crew calling us 'in sight'. Having a feathered right engine required our aircraft to need a five degree bank toward the running engine (left) causing our right wing to be further up in the air which possibly resulted in limited traffic scan ability to the right; level and below. Because the actual chain of events of all parties leading up to the near miss are not known no corrective actions can be recommended other than periodically dipping the wing of a failed engine to provide increased traffic scan ability to level and below during training/maneuvering flight. PA44 instructor and student report a NMAC with a Gulfsteam jet near OKC at 4;500 FT; while practicing engine out maneuvers. The southbound Gulfsteam traffic is reported at 12 o'clock by ATC and the PA44 pilots elect to make a 180 degree turn from east to a westerly heading. One minute later the Gulfsteam is sighted diving below the PA44 from right to left.